A339021 Number of pairs of divisors of 2n, (d1,d2), such that d1 < d2 and at least one of d1*d2 +- 1 is prime.

1, 3, 6, 6, 4, 15, 5, 9, 13, 13, 5, 26, 3, 11, 21, 12, 3, 30, 3, 22, 23, 12, 3, 40, 9, 10, 22, 20, 3, 52, 3, 14, 23, 11, 19, 51, 4, 8, 21, 32, 5, 55, 3, 20, 47, 7, 1, 53, 10, 27, 18, 16, 4, 49, 12, 32, 15, 8, 2, 92, 1, 6, 48, 17, 11, 55, 2, 20, 16, 46, 2, 79, 3, 10, 42, 18, 14, 53, 4, 42

Offset

1,2

Comments

a(n) >= 1 since for all n >= 1, 2n has the divisor pair (1,2) with 1 < 2 and 1*2 + 1 = 3 (prime).

Links

Table of n, a(n) for n=1..80.

Formula

a(n) = Sum_{d1|(2*n), d2|(2*n), d1 < d2} sign(c(d1*d2 + 1) + c(d1*d2 - 1)), where c is the prime characteristic (A010051).

Example

a(3) = 6; 2*3 = 6 has 6 divisor pairs (d1,d2) such that d1 < d2 where at least one of d1*d2 +- 1 is prime: (1,2), (1,3), (1,6), (2,3), (2,6), (3,6) as (1*2 + 1) = 3, (1*3 - 1) = 2, (1*6 + 1) = 7, (2*3 - 1) = 5, (2*6 - 1) = 11, and (3*6 - 1) = 17 (all prime).

Mathematica

Table[Sum[Sum[Sign[PrimePi[i*k - 1] - PrimePi[i*k - 2] + PrimePi[i*k + 1] -  PrimePi[i*k]]*(1 - Ceiling[2 n/k] + Floor[2 n/k]) (1 - Ceiling[2 n/i] + Floor[2 n/i]), {i, k - 1}], {k, 2 n}], {n, 80}]

Crossrefs

Cf. A010051, A337982.

Sequence in context: A247685 A372367 A185735 * A197263 A074785 A225462

Adjacent sequences: A339018 A339019 A339020 * A339022 A339023 A339024

Keyword

nonn

Author

Wesley Ivan Hurt, Nov 22 2020

Status

approved